Frequency ratio: Difference between revisions

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In the context of just intonation, ratios are almost always used to label and identify intervals and chords. However, the use of ratios to identify intervals and chords in tempered scales is also common - in these cases, it is implied that the notes are in the ''approximate'' ratio indicated. For example, a common shorthand expression might be "4:6:7:9:11 chords in [[17edo]]", which really means "the chords in which the notes are in the approximate ratio of 4:6:7:9:11 in 17edo".

== ~~Conversion~~ ==

== Conversions ==

=== ~~Cents~~ to ~~ratio~~ ===

~~To find~~ the ratio ~~''c'' for~~ an ~~interval of ''s'~~' ~~[[cent]]s~~, ~~apply~~

=== Ratio to monzo ===

Factor both the numerator and the denominator into primes. Express the entire ratio as a product of primes, each raised to an exponent. For primes appearing in the denominator, these exponents will be negative. (A prime never appears in both the numerator and the denominator.) Arrange the primes in ascending order. Enter the exponents into the monzo.

<math>\displaystyle 15/8 = (3 \cdot 5) / (2 \cdot 2 \cdot 2) = 2^{-3} \cdot 3^{1} \cdot 5^{1} = </math> {{monzo|-3 1 1}}

If any primes smaller than the largest prime don't appear, include them using a zero exponent.

<math>\displaystyle c = 2^{~~s/1200~~}</math>

<math>\displaystyle 28/27 = (2 \cdot 2 \cdot 7) / (3 \cdot 3 \cdot 3) = 2^{2} \cdot 3^{-3} \cdot 7^{1} = 2^{2} \cdot 3^{-3} \cdot 5^{0} \cdot 7^{1} = </math> {{monzo|2 -3 0 1}}

=== Monzo to ratio ===

To find the ratio ''c'' for an interval of [[monzo]] '''m''' = {{monzo| m<~~sub~~>1</~~sub~~> m<~~sub~~>2</~~sub~~> m<~~sub~~>3</~~sub~~> ~~… }}~~, apply

To find the ratio '''r''' for an interval of [[monzo]] '''m''' = {{monzo| a b c … }}, apply

<math>\displaystyle r = 2^{a} \cdot 3^{b} \cdot 5^{c} \ldots </math>

=== Ratio to cents ===

To find the cents '''c''' for a ratio '''r''', apply

<math>\displaystyle c = 1200 \cdot log(r) / log(2)</math>

Logarithms of any base (base 10, base 2, base ''e'', etc.) can be used.

<math>\displaystyle c = 1200 \cdot ln(r) / ln(2)</math>

=== Cents to ratio ===

To find the ratio '''r''' for an interval of '''c''' [[cent]]s, apply

<math>\displaystyle r = 2^{c/1200}</math>

~~<math>\displaystyle~~ c ~~= 2^{m_1} \cdot~~ 3~~^{m_2} \cdot 5^{m_3} \ldots <~~/~~math>~~

The result will be in decimal form, and will only be as exact as '''c''' is. For example, 702 cents yields 1.500038989.., which is approximately 1.5, which is 3/2.

== Extended frequency ratio (EFR) ==

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=== Subharmonic EFR, a.k.a. SEFR ===

Consider the [[melodic inversion]] of 4:5:6:7. The ratio list is 1/1, 7/6, 7/5 and 7/4, a min7flat5 chord. The EFR is 60:70:84:105. These large numbers make the chord seem more complex than it actually is because while it occurs quite high in the harmonic series, it occurs quite low in the [[subharmonic series]] as subharmonics 7, 6, 5 and 4. The subharmonic EFR or SEFR represents it as reciprocals of harmonics, in this case 1/(7:6:5:4). This list of subharmonics is spoken as "one over seven six five four". ~~Confusingly, some~~ writers ~~tend to~~ omit the numerator in SEFRs. For example, 1/(7:6:5:4) is written as 7:6:5:4.

Consider the [[melodic inversion]] of 4:5:6:7. The ratio list is 1/1, 7/6, 7/5 and 7/4, a min7flat5 chord. The EFR is 60:70:84:105. These large numbers make the chord seem more complex than it actually is because while it occurs quite high in the harmonic series, it occurs quite low in the [[subharmonic series]] as subharmonics 7, 6, 5 and 4. The subharmonic EFR or SEFR represents it as reciprocals of harmonics, in this case 1/(7:6:5:4). This list of subharmonics is spoken as "one over seven six five four". Some writers omit the numerator in SEFRs. For example, 1/(7:6:5:4) is written as 7:6:5:4.

To convert an SEFR to a ratio list, simply replace the numerator with the first subharmonic number, and put it inside each subharmonic. For example, 1/(7:6:5:4) is 7/7–7/6–7/5–7/4, with 7/7 simplifying to 1/1.

@@ Line 19: / Line 19: @@
 In the context of just intonation, ratios are almost always used to label and identify intervals and chords. However, the use of ratios to identify intervals and chords in tempered scales is also common - in these cases, it is implied that the notes are in the ''approximate'' ratio indicated. For example, a common shorthand expression might be "4:6:7:9:11 chords in [[17edo]]", which really means "the chords in which the notes are in the approximate ratio of 4:6:7:9:11 in 17edo".
-== Conversion ==
+== Conversions ==
-=== Cents to ratio ===
-To find the ratio ''c'' for an interval of ''s'' [[cent]]s, apply
+=== Ratio to monzo ===
+Factor both the numerator and the denominator into primes. Express the entire ratio as a product of primes, each raised to an exponent. For primes appearing in the denominator, these exponents will be negative. (A prime never appears in both the numerator and the denominator.) Arrange the primes in ascending order. Enter the exponents into the monzo.
+<math>\displaystyle 15/8 = (3 \cdot 5) / (2 \cdot 2 \cdot 2) = 2^{-3} \cdot 3^{1} \cdot 5^{1} = </math> {{monzo|-3 1 1}}
+If any primes smaller than the largest prime don't appear, include them using a zero exponent.
-<math>\displaystyle c = 2^{s/1200}</math>
+<math>\displaystyle 28/27 = (2 \cdot 2 \cdot 7) / (3 \cdot 3 \cdot 3) =  2^{2} \cdot 3^{-3} \cdot 7^{1} = 2^{2} \cdot 3^{-3} \cdot 5^{0} \cdot 7^{1} = </math> {{monzo|2 -3 0 1}}
 === Monzo to ratio ===
-To find the ratio ''c'' for an interval of [[monzo]] '''m''' = {{monzo| m<sub>1</sub> m<sub>2</sub> m<sub>3</sub> … }}, apply
+To find the ratio '''r''' for an interval of [[monzo]] '''m''' = {{monzo| a b c … }}, apply
+<math>\displaystyle r = 2^{a} \cdot 3^{b} \cdot 5^{c} \ldots </math>
+=== Ratio to cents ===
+To find the cents '''c''' for a ratio '''r''', apply
+<math>\displaystyle c = 1200 \cdot log(r) / log(2)</math>
+Logarithms of any base (base 10, base 2, base ''e'', etc.) can be used.
+<math>\displaystyle c = 1200 \cdot ln(r) / ln(2)</math>
+=== Cents to ratio ===
+To find the ratio '''r''' for an interval of '''c''' [[cent]]s, apply
+<math>\displaystyle r = 2^{c/1200}</math>
-<math>\displaystyle c = 2^{m_1} \cdot 3^{m_2} \cdot 5^{m_3} \ldots </math>
+The result will be in decimal form, and will only be as exact as '''c''' is. For example, 702 cents yields 1.500038989.., which is approximately 1.5, which is 3/2.
 == Extended frequency ratio (EFR) ==
@@ Line 45: / Line 66: @@
 === Subharmonic EFR, a.k.a. SEFR ===
-Consider the [[melodic inversion]] of 4:5:6:7. The ratio list is 1/1, 7/6, 7/5 and 7/4, a min7flat5 chord. The EFR is 60:70:84:105. These large numbers make the chord seem more complex than it actually is because while it occurs quite high in the harmonic series, it occurs quite low in the [[subharmonic series]] as subharmonics 7, 6, 5 and 4. The subharmonic EFR or SEFR represents it as reciprocals of harmonics, in this case 1/(7:6:5:4). This list of subharmonics is spoken as "one over seven six five four". Confusingly, some writers tend to omit the numerator in SEFRs. For example, 1/(7:6:5:4) is written as 7:6:5:4.
+Consider the [[melodic inversion]] of 4:5:6:7. The ratio list is 1/1, 7/6, 7/5 and 7/4, a min7flat5 chord. The EFR is 60:70:84:105. These large numbers make the chord seem more complex than it actually is because while it occurs quite high in the harmonic series, it occurs quite low in the [[subharmonic series]] as subharmonics 7, 6, 5 and 4. The subharmonic EFR or SEFR represents it as reciprocals of harmonics, in this case 1/(7:6:5:4). This list of subharmonics is spoken as "one over seven six five four". Some writers omit the numerator in SEFRs. For example, 1/(7:6:5:4) is written as 7:6:5:4.
 To convert an SEFR to a ratio list, simply replace the numerator with the first subharmonic number, and put it inside each subharmonic. For example, 1/(7:6:5:4) is 7/7–7/6–7/5–7/4, with 7/7 simplifying to 1/1.